In the setting of the next lemma, a standard sub-Gaussian maximal inequality would bound the largest deviation of \(Z_i\)  from its mean as \(\mathbb {E}\left[\max _{i\in [k]}| Z_i - \mu _i| \right] \leqslant (\max _{i\in [k] } \sigma _i) \sqrt{2 \log (n)}\) . For our purposes, the next result offers a critical improvement because it depends only on the variance at the likely realizations of \(I\) . A second improvement, which is the focus of the discussion in [1]}, is that the mutual information term \(I\left( Z; I \right)\)  could be much smaller than \(\log (n)\) .
